Yield to maturity as a single-rate summary

What you will be able to do

After this lesson, you should be able to:

• interpret yield to maturity as a model-dependent single-rate summary;
• state its annualization and compounding convention;
• calculate the price from an input yield without changing the promised cash flows.

What yield means in this toy model

Use the bond-payment-index $\explain{bond-payment-index}{k}$ to select one of the number-of-bond-payments $\explain{number-of-bond-payments}{n}$, and write its promised bond-cash-flow as $\explain{bond-cash-flow}{CF_{\explain{bond-payment-index}{k}}^{\mathrm{bond}}}$. Write the bond-price as $\explain{bond-price}{P_0}$ and the bond-payment-frequency as $\explain{bond-payment-frequency}{m_{\mathrm B}}$.

Write yield-to-maturity as $\explain{yield-to-maturity}{y^{(\explain{bond-payment-frequency}{m}_{\mathrm B})}}$. It is the one nominal annual rate, compounded at the bond payment frequency, that makes the discounted promised payments equal the bond price:

$$\explain{bond-price}{P_0} =\sum_{\explain{bond-payment-index}{k}=1}^{\explain{number-of-bond-payments}{n}} \frac{\explain{bond-cash-flow}{CF_{\explain{bond-payment-index}{k}}^{\mathrm{bond}}}} {\left(1+\explain{yield-to-maturity}{y}^{\left(\explain{bond-payment-frequency}{m_{\mathrm B}}\right)}/\explain{bond-payment-frequency}{m_{\mathrm B}}\right)^{\explain{bond-payment-index}{k}}}$$

(2.3.1)

The symbol $\explain{yield-to-maturity}{y^{(\explain{bond-payment-frequency}{m}_{\mathrm B})}}$ could be mistaken for a spot rate, an effective annual rate, an expected return, or another quote. In this course it means only the bond yield defined above.

Coupon rate and yield have different roles

• The coupon rate determines the promised coupon-payment $\explain{coupon-payment}{C}$ in the contract.
• The input yield determines the denominators in the price formula.

A change of the input yield does not change the coupons or the face-value $\explain{face-value}{F}$ of the bond.

Example 2.3.1 Pricing from a yield to maturity

Open the set, then choose a yield convention.

Annual payments

A two-year bond with an annual coupon rate of 5% and a face value of USD 1,000 pays USD 50 and USD 1,050. At an annual yield of 6% with annual compounding, the price is:

$$\explain{bond-price}{P_0} =\frac{50}{1.06} +\frac{1050}{1.06^2} \approx\text{USD }981.67$$

Semiannual compounding

For the same face value and coupon rate over two years, but with semiannual payments, $\explain{coupon-payment}{C}=25$, $\explain{bond-payment-frequency}{m_{\mathrm B}}=2$, and $\explain{number-of-bond-payments}{n}=4$. A nominal annual yield of 6% gives a periodic rate of 3%, and the price is:

$$\explain{bond-price}{P_0} =\frac{25}{1.03} +\frac{25}{1.03^2} +\frac{25}{1.03^3} +\frac{1025}{1.03^4} \approx\text{USD }981.41$$

The annual and the semiannual example use the same yield number, 6%, but different payment times and different compounding.

Zero-coupon special case

With no coupons and maturity-time $\explain{maturity-time}{T}$, the price is:

$$\explain{bond-price}{P_0}=\frac{\explain{face-value}{F}}{\left(1+\explain{yield-to-maturity}{y}^{\left(\explain{bond-payment-frequency}{m_{\mathrm B}}\right)}/\explain{bond-payment-frequency}{m_{\mathrm B}}\right)^{\explain{bond-payment-frequency}{m_{\mathrm B}}\explain{maturity-time}{T}}}$$

For USD 1,000 due in three years at 4% with annual compounding, the price is approximately USD 889.00.

Check your understanding

These items separate yield interpretation from annual and periodic-compounding calculations. Each question stays collapsed until you open it; answers and explanations appear once you check.

Knowledge check 2.3.1 Yield to maturity

1. DirectQuestion 1Not answered

2. TransferQuestion 2Not answered

3. DirectQuestion 3Not answered

4. TransferQuestion 4Not answered

Model boundary and review note

This lesson calculates the price from an input yield. It does not solve numerically for the yield from a price. A real bond implementation also needs schedules, day counts, accrued interest, clean and dirty prices, and possibly credit and option models.

The single-rate definition and the $2\explain{maturity-time}{T}$-period pricing formula follow Tuckman & Serrat[1]; Hull gives the same definition, solved iteratively[2]. FINRA defines it as the return to an investor who buys at the market price and holds to maturity[3]. The lesson stays draft pending human confirmation of the printed locators.

References

1. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §3.2, “Yield to Maturity”: the single rate that discounts a bond’s cash flows to its market price (eq. 3.5 and the general formula), also its internal rate of return. draft ↩
2. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). Ch. 4, “Bond Yield”. draft ↩
3. FINRA, Understanding Bond Yield and Return. “Yield to maturity” section. https://www.finra.org/investors/insights/bond-yield-return draft ↩